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Proving a subspace

  1. Sep 15, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that the set of all n x n matrices A such that AB = BA for a fixed n x n matrix B, is a subspace of Mnn.

    2. Relevant equations

    u + v is in the same vector space as u and v.
    ku is in the same vector space as u, where k is any real number.

    3. The attempt at a solution

    I am drawn to think of a diagonal matrix when I think of this question. And if I multiply a diagonal matrix by a scalar, it can only be a diagonal matrix or the zero matrix, either way, it AB still equals BA. Similarly, adding two diagonal matrices obtains either another diagonal matrix or the zero matrix...so, in this way, A is a subspace of Mnn.

    Am I correct?
     
  2. jcsd
  3. Sep 15, 2011 #2

    lanedance

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    Homework Helper

    i would just test the subspace requirements directly:

    clearly the zero matrix is n the set
    now say A,B satisfy AB = BA then is cA+dB in the set? that will satisfy closure under scalar multiplication and addition
     
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